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Vol. 14 (2009), Paper no. 57, pages 1705–1726.
Journal URL
http://www.math.washington.edu/~ejpecp/
Spontaneous breaking of continuous rotational symmetry
in two dimensions
Franz Merkl
Mathematical Institute
University of Munich
Theresienstr. 39
D-80333 Munich, Germany
Email: merkl@mathematik.uni-muenchen.de
Silke W.W. Rolles
Zentrum Mathematik, Bereich M5
Technische Universität München
D-85747 Garching bei München, Germany
Email: srolles@ma.tum.de
Abstract
In this article, we consider a simple model in equilibrium statistical mechanics for a twodimensional crystal without defects. In this model, the local specifications for infinite-volume
Gibbs measures are rotationally symmetric. We show that at sufficiently low, but positive temperature, rotational symmetry is spontaneously broken in some infinite-volume Gibbs measures.
Key words: Gibbs measure, rotation, spontaneous symmetry breaking, continuous symmetry.
AMS 2000 Subject Classification: Primary 60K35; Secondary: 82B20, 82B21.
Submitted to EJP on June 21, 2007, final version accepted July 1, 2009.
1705
1 Introduction
According to the famous Mermin-Wagner theorem and its more recent variants, Gibbs equilibrium
distributions of thermodynamic systems in two dimensions usually preserve continuous symmetries
of their local specifications. This holds, under weak conditions, for the preservation of translational
symmetry and for inner symmetries like spin rotations. Important results concerning preservation
of continuous symmetries in two dimensions have been proven by Mermin and Wagner [MW66],
Dobrushin and Shlosman [DS75], Fröhlich and Pfister [FP81], see also[FP86], Pfister [Pfi81], and
by Georgii [Geo99]. Over the decades, the regularity conditions on the interaction potential in
Mermin-Wagner type theorems have been astonishingly weakened. Ioffe, Shlosman, and Velenik
[ISV02] have proven the absence of continuous symmetry breaking in two-dimensional lattice systems without any smoothness assumptions on the interaction. Recently, a general version of this
preservation of symmetries has been shown by Richthammer in [Ric05] and [Ric09], using only
very weak regularity assumptions for the local Hamiltonians. This includes, among others, systems
of hard disks in two dimensions at any chemical potential.
On the other hand, theorems of Mermin-Wagner type [MW66] are not applicable to spatial rotations
in two dimensions. For example, it is conjectured that not all Gibbsian equilibrium distributions
describing hard disks in two dimensions at high chemical potentials are rotationally symmetric.
Already Fröhlich and Pfister [FP81] remarked that
(. . . ) the breaking of rotation-invariance, i.e. directional ordering, is possible, in principle, in two-dimensional systems with connected correlations which do not fall off more
rapidly than the inverse square distance (. . . )
In this paper, it will be shown that some infinite-volume Gibbsian lattice particle systems in two
dimensions indeed show spontaneous breaking of spatial rotational symmetry at low temperature.
This symmetry breaking is shown for a system of two-dimensional “atoms”. In this system, the atoms
are characterized by their location in the plane and their orientation. The local Hamiltonians are
chosen rotationally symmetric. Nevertheless, the particle configurations are indexed by a triangular
lattice. Thus, the permitted particle configurations may be viewed as a deformed triangular lattice.
Motivation. In the physics literature, spontaneous breaking of rotational symmetry in twodimensional solids has been taken as a fact already a long time ago; see e.g. [Mer68] in the case
of a two-dimensional harmonic net. Even more, in the physics literature, melting transitions of
two-dimensional solids and liquid and “hexatic” phases have been discussed, see e.g. [NH79].
However, from a fundamental, statistical mechanical point of view, the difference between crystalline solids and fluids is not well understood rigorously. The remarkable recent paper [BLRW06]
shows the existence of a solidification phase transition for interacting tiles with zippers. With the
exception of this reference, we are not aware of any other proof of a melting/freezing phase transition in a continuum system of interacting particles in thermal equilibrium. Even worse, there is
no proof of crystalline order at low, but positive temperature for any realistic continuum particle
system. It seems reasonable to take the breaking of rotational symmetry as one characteristic property of crystals. A goal in the long run is to understand models for crystallization of increasing
complexity from a fundamental point of view. As a starting point, we examine a simplified model
for a two-dimensional crystal. We exclude defects of the crystal by definition of the model, taking
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an infinite system of particles that looks locally like a slightly perturbed triangular lattice, up to
perturbations of size ǫ > 0. Intuitively, one may think of ǫ being small, although this is not required
in our model. The only reason why we work with the triangular lattice instead of square lattices is
that we think that the triangular lattice may be a more realistic model for ordered two-dimensional
particle systems. Our proof of spontaneous breaking of rotational symmetry in this model relies on
a simple variant of infrared bounds; no reflection positivity is required.
The goal of this paper is to understand some features of two-dimensional crystals at low temperature
in a simplified model. It tells us nothing about melting/freezing phase transitions. We do not know
the high-temperature behaviour of the model discussed in this paper. It may be possible that for
small ǫ > 0, it shows spontaneous breaking of rotational symmetry at all temperatures, in other
words, it might have no melting transition. One may compare it with the following: For a spin
model with O(2)-invariant ferromagnetic interaction with a constraint that enforces approximate
local alignment of the spins, Aizenman [Aiz94] proves a power-law lower bound for the correlation
function at all temperatures. Similar to the Patrascioiu-Seiler model studied by Aizenman [Aiz94],
our model forbids topological defects by using appropriate constraints. The present model also
forbids lattice defects. It is a major task for future work to remove these constraints.
2 Results
Definition of the model. Let τ := eπi/3 , and consider the triangular lattice T = (V, E) with vertex
set V := Z + τZ and set of directed edges E := {(u, v) : u, v ∈ V with |u − v| = 1}. In particular,
there are two directed edges with opposite directions between any pair of vertices at distance one
from each other.
We consider an infinite system of atoms, indexed by the vertex set V of the triangular lattice. Intuitively speaking, every atom has six “arms”. At the end of each arm, there is a “sticky ball” of radius
ǫ, where ǫ > 0 is fixed (see Figure 1).
Figure 1: An atom consisting of 6 arms with sticky balls at the ends of the arms.
ǫ
At each of the six sticky balls, another atom is located. The neighboring atom is located somewhere
inside the ball of radius ǫ; it has again six sticky arms. In Figure 2, three atoms sticking together are
shown.
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Figure 2: A molecule consisting of three atoms.
Formally, for Λ ⊆ V , define the space ΩΛ of configurations on Λ by
ΩΛ := {(x u , γu )u∈Λ ∈ (C × S 1 )Λ : x 0 = 0}
(2.1)
if 0 ∈ Λ, and ΩΛ := (C × S 1 )Λ otherwise. Thus, a configuration consists of the position x u ∈ C of the
atom with index u and its orientation γu ∈ S 1 = {z ∈ C : |z| = 1} for every u ∈ Λ. The condition
x 0 = 0 is a reference fixation of the configurations. We write
γu = e iαu
with
− π < αu ≤ π.
(2.2)
A configuration in ΩΛ is of the form
ωΛ = (ωu )u∈Λ
with
ωu := (x u , γu ).
(2.3)
For disjoint Λ, Λ′ ⊆ V , set ωΛ ωΛ′ := ωΛ ∪ ωΛ′ . We endow ΩΛ with the σ-field FΛ generated by the
projections ωΛ 7→ x u and ωΛ 7→ γu to the individual coordinates and the reference measure
¨
Q
Q
δ0 (d x 0 ) u∈Λ\{0} d x u v∈Λ dγ v for 0 ∈ Λ,
Q
(2.4)
ρΛ (dωΛ ) := Q
otherwise;
v∈Λ dγ v
u∈Λ d x u
here δ0 denotes the unit point mass in 0, and d x u and dγ v denote the Lebesgue measure on C and
S 1 , respectively.
Let ǫ > 0 be a fixed perturbation parameter, and let h : C → [0, ∞] be a potential with the following
properties:
• h is measurable and Lebesgue-almost everywhere continuous.
• h is rotationally symmetric: h(a) = h(b) for |a| = |b|.
• For all a ∈ C,
h(a) ≥ |a|2 + ∞ · 1{|a|≥ǫ} .
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(2.5)
• Furthermore,
c1 (ǫ ′ ) :=
sup
h(a)
(2.6)
a∈C:|a|<ǫ ′
satisfies
c1 (ǫ) < ∞
and
lim c1 (ǫ ′ ) = 0.
ǫ ′ →0
(2.7)
One possible choice for the potential is the function h(a) = |a|2 + ∞ · 1{|a|≥ǫ} .
Let Λ ⊂ V be a finite set containing 0 with outer boundary ∂ Λ and set Λ := Λ ∪ ∂ Λ. For ωΛ ∈ ΩΛ
and ω∂ Λ ∈ Ω∂ Λ with ωΛ ω∂ Λ = (x u , γu )u∈Λ , define the local Hamiltonian with boundary condition
ω∂ Λ by
X
HΛ (ωΛ |ω∂ Λ ) :=
h(x u − x v − γu (u − v)).
(2.8)
u,v∈Λ: (u,v)∈E
The partition sum associated with this local Hamiltonian at the inverse temperature β > 0 is given
by
Z
e−β HΛ (ωΛ |ω∂ Λ ) ρΛ (dωΛ )
Zβ,Λ (ω∂ Λ ) :=
(2.9)
ΩΛ
with the convention e−∞ = 0. Note that Zβ,Λ takes finite values.
Clearly, for ωΛ ω∂ Λ = (x u = u, γu = 1)u∈Λ , the local Hamiltonian is finite. This configuration has
an atom with “standard orientation” at every u ∈ Λ; i.e. the atoms are located at the vertices of the
triangular lattice. All other configurations with finite energy may be viewed as perturbations of this
triangular lattice configuration.
For ω∂ Λ ∈ Ω∂ Λ such that Zβ,Λ (ω∂ Λ ) > 0, the finite-volume Gibbs measure at the temperature T =
1/β with the boundary condition ω∂ Λ is defined to be the following probability measure on ΩΛ :
µβ,Λ (dωΛ |ω∂ Λ ) :=
e−β HΛ (ωΛ |ω∂ Λ )
Zβ,Λ (ω∂ Λ )
ρΛ (dωΛ ).
(2.10)
A probability measure µβ on (ΩV , FV ) is called an infinite-volume Gibbs measure at inverse temperature β, if it satisfies the DLR-conditions:
Z
1
µβ (A|ωΛc ) =
1A(ω̃Λ ωΛc )e−β HΛ (ω̃Λ |ω∂ Λ ) ρΛ (d ω̃Λ ) µβ -a.s.
(2.11)
Zβ,Λ (ω∂ Λ ) Ω
Λ
for any A ∈ FV and any finite Λ ⊂ V containing 0. In particular, this includes the requirement
Zβ,Λ (ω∂ Λ ) > 0 for µβ -almost every ω ∈ ΩV .
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Rotational symmetry of the local specifications. For s ∈ S 1 and any Λ ⊆ V , define the rotation
Rs : ΩΛ → ΩΛ , (x u , γu )u∈Λ 7→ (s x u , sγu )u∈Λ . For finite Λ ⊂ V with 0 ∈ Λ, note that both, the reference
measures ρΛ and the local Hamiltonians, are invariant with respect to Rs :
HΛ (Rs ωΛ |Rs ω∂ Λ ) = HΛ (ωΛ |ω∂ Λ ),
(ωΛ ∈ ΩΛ , ω∂ Λ ∈ Ω∂ Λ ).
(2.12)
An infinite-volume Gibbs measure µβ is called rotationally invariant, if for all s ∈ S 1 , the image
measure Rs [µβ ] of µβ under the map Rs equals µβ . Otherwise, we say that µβ spontaneously breaks
the rotational symmetry.
Theorem 2.1 (Spontaneous breaking of rotational symmetry). Let ǫ > 0. At all sufficiently small
temperatures β −1 > 0, there exists an infinite-volume Gibbs measure µβ that spontaneously breaks the
rotational symmetry. More precisely,
Eµβ [α0 ] = 0
and
lim Varµβ (α0 ) = 0,
β→∞
(2.13)
so that for all sufficiently low temperatures β −1 , with respect to µβ , the orientation γ0 of the 0-th
particle cannot be uniformly distributed on S 1 .
Nevertheless, there is a homotopy between the standard configuration and a configuration that
equals the standard configuration near infinity, but is rotated by an arbitrarily large angle on any
fixed finite piece. The following remark states this more formally. Its proof is sketched at the end of
Section 4.
Remark 2.2 (Arbitrary rotations of finite pieces). For every ǫ > 0, every finite set Λ ⊂ V , and every
ϕ > 0, there is a path (x u (t), γu (t))u∈V,−ϕ≤t≤ϕ in ΩV with the following properties:
(a) Standard configuration at deformation parameter t = 0: For all u ∈ V , x u (0) = u and γu (0) = 1.
(b) Admissible configurations: For all t ∈ [−ϕ, ϕ], the configuration (x u (t), γu (t))u∈V is admissible, i.e. for all (u, v) ∈ E, the bound |x u − x v − γu (u − v)| < ǫ holds.
(c) Continuity: For all u ∈ V , the map t 7→ (x u (t), γu (t)) is continuous.
(d) Rotation on Λ: For all u ∈ Λ and t ∈ [−ϕ, ϕ], we have x u (t) = e i t u and γu (t) = e i t .
(e) Standard configuration near ∞: There is a finite set Σ with Λ ⊆ Σ ⊂ V , such that for all
u ∈ V \ Σ and all t ∈ [−ϕ, ϕ], one has x u (t) = u and γu (t) = 1.
Finite-volume Gibbs measures with periodic boundary conditions. For N ∈ N, let ΛN be a set
of representatives for (Z + τZ)/(N Z + N τZ) with 0 ∈ ΛN . More specifically, we take
ΛN = {k + τl : k, l ∈ {⌊−N /2⌋, . . . , ⌊N /2⌋ − 1}}.
We introduce the space of all spatially periodic configurations with respect to N :
(x u , γu )u∈V ∈ (C × S 1 )V : x 0 = 0, x u+l = x u + l, γu+l = γu for all
per
ΩN :=
.
u ∈ V, l ∈ NV
1710
(2.14)
(2.15)
Note that the average density of particles for these configurations is just the same as in the standard
triangular lattice V , due to the condition x u+l = x u + l rather than x u+l = x u + const · l. Sometimes,
per
per
it is convenient to identify ΩN with the set ΩΛN = {ωΛN : ω ∈ ΩN }. We will implicitly use this
identification below.
For periodic boundary conditions with period N , we define the finite-volume Hamiltonian H N in
analogy to (2.8)
X X
per
h(x u − x v − γu (u − v)),
ω = (x u , γu )u∈V ∈ ΩN .
(2.16)
H N (ω) :=
u∈ΛN
v∈V :
(u,v)∈E
Note that this definition does not depend on the choice of the set ΛN of representatives for V /N V .
sqr
In the special case h(a) = |a|2 + ∞ · 1{|a|≥ǫ} , we denote this function H N by H N .
Furthermore, we define the finite-volume Gibbs measure at inverse temperature β with periodic
per
boundary conditions to be the following probability measure on ΩN :
µβ,N (dω) :=
e−β H N (ω)
Zβ,N
ρN (dω)
(2.17)
with the partition sum
Zβ,N :=
Z
per
ΩN
e−β H N dρN > 0.
per
(2.18)
per
Here, ρN denotes the reference measure on ΩN identified with ρΛN via the identification ΩN ≡
ΩΛ N .
The following theorem is a finite-volume analogue of Theorem 2.1. The given bound is uniform in
the size of the finite box.
Theorem 2.3 (Finite-volume estimate). For all ǫ > 0 and all δ > 0, there exist β0 > 0 and N0 ∈ N
such that for all β ∈ (β0 , ∞) and all N ≥ N0 , the variance of α0 with respect to the Gibbs measure µβ,N
is bounded above by δ:
Varµβ,N (α0 ) < δ.
(2.19)
Theorems 2.1 and 2.3 hold also for slightly different local Hamiltonians. For instance, we could
index the atoms by the square lattice instead of the triangular lattice, and our results still hold.
The following theorem provides the link between the infinite-volume result in Theorem 2.1 and the
finite-volume estimate in Theorem 2.3.
Theorem 2.4 (Infinite-volume limit). For all ǫ > 0 and all β > 0, there exists a strictly increasing
sequence (Nk )k∈N of natural numbers such that the finite-dimensional marginals of (µβ,Nk )k∈N converge
weakly to the finite-dimensional marginals of an infinite-volume Gibbs measure µβ as specified by the
DLR-conditions (2.11).
In particular, this means that for any (u, v) ∈ E, the inequality |x u − x v − γu (u − v)| < ǫ holds µβ almost surely. Furthermore, all finite-dimensional marginals of µβ are absolutely continuous with
respect to the corresponding reference measures.
1711
Intuitive ideas behind the proofs. Before we prove our results, let us explain some intuitive
ideas in a simpler model. This paragraph serves only to provide some rough intuitive background
information without giving detailed proofs. The formal proofs of our theorems in later sections can
be checked even if one ignores the intuitive picture.
0
Let x u0 := u, α0u := 0, and γ0u := e iαu = 1 denote the coordinates of the “standard configuration” with
one particle with standard orientation at every vertex u. We set yu := x u − x u0 = x u − u. Note that
αu = αu − α0u . Let ⟨w, z⟩ = Re(wz̄) denote the Euclidean scalar product of w, z ∈ C.
sqr
Let us consider the second order Taylor approximation at x 0 and α0 of the Hamiltonian H N viewed
as a function of x and α. In these variables, the Hamiltonian of the linearized model is given by
X X ¦
©
gauss
| yu − y v |2 − 2⟨ yu − y v , i(u − v)⟩αu + α2u
(2.20)
H N ( y, α) :=
u∈ΛN
v∈V :
(u,v)∈E
for N -periodic configurations ( y, α) = ( yu , αu )u∈V ∈ (C×R)V . The corresponding local Hamiltonians
for the infinite-volume system are invariant with respect to the linearized rotations
R̃ t : ( yu , αu )u 7→ ( yu + t iu, αu + t)u
(2.21)
for all t ∈ R. However, since these linearized rotations do not preserve periodic boundary conditions,
they are no symmetries of the model in finite volume with periodic boundary conditions.
Let
V ∗ := {p ∈ C : ⟨p, v⟩ ∈ 2πZ for all v ∈ V } = τ∗1 Z + τ∗2 Z,
(2.22)
p
p
where τ∗1 = 2π(1 − i/ 3) and τ∗2 = 4πi/ 3, denote the dual lattice to V . Furthermore, let Λ∗N be a
set of representatives for (N −1 V ∗ ) mod V ∗ with 0 ∈ Λ∗N . For example, one may take
Λ∗N = {N −1 m1 τ∗1 + N −1 m2 τ∗2 : m1 , m2 ∈ {0, . . . , N − 1}}.
We introduce Fourier variables:
X
yu e−i⟨p,u⟩
ŷ p = |ΛN |−1
and
u∈ΛN
α̂ p = |ΛN |−1
X
αu e−i⟨p,u⟩
(2.23)
(2.24)
u∈ΛN
with the inverse
yu =
X
ŷ p e i⟨p,u⟩
and
p∈Λ∗N
αu =
X
α̂ p e i⟨p,u⟩ .
(2.25)
p∈Λ∗N
gauss
Because of translational symmetry, the Fourier-transform block-diagonalizes the Hamiltonian H N :
Define the matrix A p for all p ∈ C by
Œ
‚
X
|1 − e i⟨p,l⟩ |2 l(1 − e−i⟨p,l⟩ )
,
(2.26)
Ap =
l(1 − e i⟨p,l⟩ ) 1
l∈N
where N := {l ∈ V : |l| = 1} denotes the set containing the six neighbors of the origin in the
triangular lattice. Then, one has for all N -periodic ( y, α):
‚
Œ
X
ŷ p
gauss
.
(2.27)
H N ( y, α) = |ΛN |
( ŷ p , i α̂ p )A p
i α̂ p
∗
p∈ΛN
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The only points where A p is non-invertible, are p ≡ 0 mod V ∗ ; see (3.20), below. Close to the
singularity, i.e. in the infrared regime p → 0, one has
‚
Œ
|p|2 ip
Ap = 3
+ lower order terms,
(2.28)
−ip 2
and consequently,
A−1
p
=
1
3
‚
2|p|−2 −ip|p|−2
ip|p|−2 1
Œ
+ lower order terms.
(2.29)
Consequently, if we denote the entries of a 2 × 2-matrix B by B j,k , j, k = 1, 2, we obtain
( ŷ p , i α̂ p )A p
‚
ŷ p
i α̂ p
Œ
≥
det A p
|α̂ p |2 =
(A p )1,1
|α̂ p |2
(A−1
p )2,2
≥ c2 |α̂ p |2
(2.30)
for all p close to 0 with a constant c2 > 0. If p stays bounded away from the singularity p ≡
0 mod V ∗ , the same bound holds. Hence, unlike the well-known infrared bounds [FSS76] for the
classical isotropic Heisenberg model in three or more dimensions, where the two-point function in
Fourier space is bounded by O(|p|−2 ) as p → 0, the relevant entry of the covariance matrix in the
linearized model is bounded by a constant, uniformly in p. This behavior remains the same for the
model with Hamiltonian H N . The uniform lower bound for the finite-volume Hamiltonians H N is a
crucial ingredient of our proofs. We derive it in Section 3. From this bound, we deduce the estimate
for the variance of the angle α0 with respect to µβ,N stated in Theorem 2.3. In Section 4, we show
how to pass to the infinite-volume limit.
Remark on the one-dimensional analogue of this model. We emphasize that the mechanism
for spontaneous breaking of rotational symmetry as described above in the linearized model works
only in dimension two (or higher dimensions). In one dimension, the linearized model does not
exhibit spontaneous breaking of rotational symmetry. Consider a one-dimensional chain of twoarmed molecules in a plane. One obtains the corresponding linearized model from (2.20) by replacing the two-dimensional vertex set V by the one-dimensional line Z with nearest-neighbour
edges and replacing ΛN by {0, . . . , N − 1}. This model has a completely different behavior than
its two-dimensional version. Let us sketch why. The representation (2.27/2.26) of the linearized
Hamiltonian remains valid with Λ∗N = {2π j/N : j = 0, 1, . . . , N − 1} and N = {−1, +1}. However,
in the one-dimensional model,
Œ
‚
(2 sin(p/2))2 i sin p
(2.31)
Ap = 2
−i sin p
1
which implies
(A−1
p )2,2 =
1
2 sin2 (p/2)
=
2
p2
+ O(1)
as p → 0.
(2.32)
In contrast to the two-dimensional model, this is not integrable near 0. As a consequence, the symmetry of linearized rotations is preserved in the corresponding infinite-volume model. This means
1713
that the one-dimensional linearized model does not have infinite-volume Gibbs measures unless one
pins down one directional coordinate, setting e.g. α0 = 0.
Of course, this analysis tells us nothing about the non-linearized one-dimensional model in a rigorous sense. However, it indicates that if rotational symmetry is broken in the one-dimensional
non-linearized model, this is caused by a completely different mechanism, not accessible in the
linearization.
3 Finite-volume arguments
The aim of this section is to prove Theorem 2.3. Throughout the remainder of the article, we fix
ǫ > 0. First, we state two simple properties of the finite-volume Gibbs measure µβ,N with periodic
boundary conditions:
Lemma 3.1 (Symmetry properties of the law of α v ). For all N ∈ N, β > 0, and v ∈ V , one has
Lawµβ,N (α v ) = Lawµβ,N (α0 )
and
Eµβ,N [α v ] = 0.
(3.1)
Proof. Let N ∈ N, β > 0, and v ∈ ΛN . The Hamiltonian H N and the reference measure ρN are
per
invariant under the translation (x u , γu )u∈V 7→ (x u+v − x v , γu+v )u∈V . Since the configurations in ΩN
are spatially periodic, the distribution µβ,N is also invariant under this translation. It follows that γ v
has the same distribution as γ0 under µβ,N . Hence, Lawµβ,N (α v ) = Lawµβ,N (α0 ).
Next, since (u, v) ∈ E iff (u, v) ∈ E and h(a) = h(a) for all a ∈ C, one obtains
X X
X X
h(x u − x v − γu (u − v))
h(x u − x v − γu (u − v)) =
u∈ΛN
v∈V :
(u,v)∈E
=
u∈ΛN
v∈V :
(u,v)∈E
X
X
u∈ΛN
v∈V :
(u,v)∈E
h(x u − x v − γu (u − v)).
(3.2)
Thus, the distribution µβ,N is invariant under the reflection (x u , γu )u∈V 7→ (x u , γu )u∈V . In particular,
γ0 and γ0 have the same law; hence α0 and −α0 also have the same law, since there is no mass in
α0 = π. Consequently, Eµβ,N [α0 ] = 0.
Note that the properties (2.5) and (2.7) of the potential imply that
per
{H N < ∞} = {ω ∈ ΩN : |x u − x v − γu (u − v)| < ǫ for all u, v ∈ V with (u, v) ∈ E}.
(3.3)
The following lemma states a crucial lower bound for the local Hamiltonian with periodic boundary
conditions.
Lemma 3.2 (Bounding the Hamiltonian). Let c3 = 8/π2 . For all N ∈ N and all ω = (x u , γu )u∈V ∈
per
ΩN , the Hamiltonian satisfies the bound
X
H N (ω) ≥ c3
α2u .
(3.4)
u∈ΛN
1714
After a Fourier transform, this inequality may be viewed as a variant of formula (2.30) for Fourier
variables p not necessarily close to 0.
Proof of Lemma 3.2. It suffices to prove the bound (3.4) on the set {H N < ∞}. Recall the definition
(2.23) of Λ∗N . We introduce again Fourier variables: For u ∈ ΛN , set
X
x u − u =: yu =
ŷ p e i⟨p,u⟩ ,
(3.5)
p∈Λ∗N
γu − 1 =: zu =
X
ẑ p e i⟨p,u⟩ .
(3.6)
p∈Λ∗N
Inserting these identities, we get for any u, v ∈ ΛN
|x u − x v − γu (u − v)|2 =|(x u − u) − (x v − v) + (γu − 1)(v − u)|2
=| yu − y v + zu (v − u)|2
X
© 2
¦
=
e i⟨p,u⟩ ŷ p (1 − e i⟨p,v−u⟩ ) + ẑ p (v − u) .
(3.7)
p∈Λ∗N
Recall that N denotes the set of neighbors of the origin in the triangular lattice. For l ∈ N , p ∈ Λ∗N ,
and u ∈ ΛN , set
ŵ p,l := ŷ p (1 − e i⟨p,l⟩ ) + ẑ p l,
X
wu,l :=
ŵ p,l e i⟨p,u⟩ .
(3.8)
(3.9)
p∈Λ∗N
Inserting these abbreviations in (3.7) yields the following expression for the Hamiltonian H N on
{H N < ∞}:
X X
sqr
|x u − x v − γu (u − v)|2
H N (ω) ≥ H N (ω) =
u∈ΛN
=
X
u∈ΛN
v∈V :
(u,v)∈E
2
X X
ŵ p,v−u e i⟨p,u⟩ .
(3.10)
v∈V :
p∈Λ∗N
(u,v)∈E
If (u, v) ∈ E, then v − u ∈ N . Using the definition of wu,l and applying Parseval’s identity, we obtain
2
X X X
sqr
H N (ω) =
ŵ p,l e i⟨p,u⟩ u∈ΛN l∈N
=
X X
l∈N u∈ΛN
p∈Λ∗N
|wu,l |2 = |ΛN |
X X
l∈N
p∈Λ∗N
|ŵ p,l |2 .
(3.11)
Next, we insert in the last expression the definition (3.8) of ŵ p,l :
X X
sqr
H N (ω) =|ΛN |
| ŷ p (1 − e i⟨p,l⟩ ) + ẑ p l|2
p∈Λ∗N l∈N
=|ΛN |
X
( ŷ p , ẑ p )A p
p∈Λ∗N
1715
‚
ŷ p
ẑ p
Œ
(3.12)
with the same positive semidefinite matrix
‚
Œ
X
|1 − e i⟨p,l⟩ |2 l(1 − e−i⟨p,l⟩ )
Ap =
l(1 − e i⟨p,l⟩ ) |l|2
(3.13)
l∈N
as in (2.26). In order to bound the quadratic form given by A p from below, we rewrite (A p )1,1 and
then derive a lower bound for det A p .
Set
2
1
p
1
p
N+ := {1, τ, τ } = 1, (1 + i 3), (−1 + i 3) .
2
2
(3.14)
Then, N = N+ ∪ (−N+ ). Note that |1 − e i⟨p,l⟩ |2 = |1 − e i⟨p,−l⟩ |2 = 2(1 − cos⟨p, l⟩) holds for l ∈ N .
Consequently,
X
X
(A p )1,1 =
|1 − e i⟨p,l⟩ |2 = 4
(1 − cos⟨p, l⟩) = 4(3 − cos s1 − cos s2 − cos s3 )
(3.15)
l∈N
l∈N+
with s1 = ⟨p, 1⟩, s2 = ⟨p, τ⟩, and s3 = ⟨p, τ2 ⟩.
For the determinant, we get:
det A p = 6(A p )1,1 − (A p )1,2 (A p )2,1 .
(3.16)
We rewrite the last term in the last equation:
2
X
l(1 − e−i⟨p,l⟩ )
(A p )1,2 (A p )2,1 =
l∈N
2
X
−i⟨p,l⟩
−i⟨p,−l⟩ l(1 − e
) − l(1 − e
)
=
l∈N+
X
2
=
2il sin⟨p, l⟩
l∈N+
2
p
p
=2 sin s1 + (1 + i 3) sin s2 + (−1 + i 3) sin s3 =(2 sin s1 + sin s2 − sin s3 )2 + 3(sin s2 + sin s3 )2 .
(3.17)
Expanding the last term and using the estimate 4x y ≤ 2(x 2 + y 2 ) yields:
(2 sin s1 + sin s2 − sin s3 )2 + 3(sin s2 + sin s3 )2
=4(sin2 s1 + sin2 s2 + sin2 s3 ) + 4 sin s2 sin s3 + 4 sin s1 sin s2 − 4 sin s1 sin s3
≤8(sin2 s1 + sin2 s2 + sin2 s3 ).
(3.18)
Observe that sin2 x = 1−cos2 x = (1+cos x)(1−cos x) ≤ 2(1−cos x) holds for all x ∈ R. Combining
(3.17) and (3.18) with this bound yields
(A p )1,2 (A p )2,1 ≤ 16(3 − cos s1 − cos s2 − cos s3 ).
1716
(3.19)
Inserting (3.15) and (3.19) into (3.16) we get:
det A p ≥ 8(3 − cos s1 − cos s2 − cos s3 ) = 2(A p )1,1 .
(3.20)
In particular, det A p and (A p )1,1 can vanish only if s1 , s2 , and s3 are integer multiples of 2π, that is,
for p ∈ V ∗ . As a consequence, for all p ∈ Λ∗N \ {0}, we have
1
(A−1
p )2,2
=
det A p
(A p )1,1
≥ 2.
(3.21)
Thus, we get the following lower bound for the quadratic form described by A p :
( ŷ p , ẑ p )A p
‚
ŷ p
ẑ p
Œ
≥ 2|ẑ p |2
(3.22)
for all p ∈ Λ∗N . For p ∈ Λ∗N \ {0}, this is an immediate consequence of (3.21). In the singular case
p = 0, the claim (3.22) is clear:
‚
Œ
ŷ0
( ŷ 0 , ẑ 0 )A0
= 6|ẑ0 |2 ≥ 2|ẑ0 |2 .
(3.23)
ẑ0
We conclude from (3.12) and Parseval’s identity that
X
X
H N (ω) ≥ 2|ΛN |
|ẑ p |2 = 2
|1 − γu |2 .
p∈Λ∗N
(3.24)
u∈ΛN
Since |1 − e iα |2 = 2(1 − cos α) ≥ 4α2 /π2 for all α ∈ (−π, π], the claim (3.4) follows with c3 =
8/π2 .
Lemma 3.3 (Bounding the partition sum). Let c3 be as in Lemma 3.2 and c1 as in (2.6). For all
δ > 0, β > 0, N ∈ N, and all ǫ ′ ∈ (0, π−1/2 ) with c1 (3ǫ ′ ) < c3 δ/48, the partition sum defined in
(2.18) satisfies the following lower bound:
Zβ,N ≥ exp({c4 − β c3 δ/8}|ΛN |)
(3.25)
with c4 = c4 (ǫ ′ ) = log(2π(ǫ ′ )3 ).
Note that by (2.7), c1 (ǫ ′ ) → 0 as ǫ ′ → 0, and consequently, there exists ǫ ′ with the required properties.
Proof of Lemma 3.3. Here is the idea: The contribution of the configurations globally close to the
standard configuration suffice to show the claimed lower bound.
Let δ > 0, β > 0, and N ∈ N. We have
Z
Zβ,N ≥
e−β H N dρN
H N ∈[0,c3 δ|ΛN |/8]
≥ exp(−β c3 δ|ΛN |/8)ρN H N ∈ [0, c3 δ|ΛN |/8] .
1717
(3.26)
Let ǫ ′ ∈ (0, π−1/2 ) with c1 (3ǫ ′ ) < c3 δ/48, and define
per
Uǫ′ := {ω ∈ ΩN : |x u − u| < ǫ ′ and |αu | < ǫ ′ ∀u ∈ V }.
(3.27)
Uǫ′ ⊆ {H N ∈ [0, c3 δ|ΛN |/8]}.
(3.28)
We claim that
To prove this, let ω = (x u , e iαu )u∈V ∈ Uǫ′ . Then, for any (u, v) ∈ E,
|x u − x v − e iαu (u − v)| ≤ |x u − u| + |v − x v | + |1 − e iαu | · |u − v| < 3ǫ ′ ;
(3.29)
here we used that |1 − e iαu | ≤ |αu | and |u − v| = 1. Consequently, using the abbreviation (2.6), the
last estimate implies
X X
h(x u − x v − e iαu (u − v)) ≤ 6|ΛN |c1 (3ǫ ′ ) < |ΛN |c3 δ/8;
(3.30)
H N (ω) =
u∈ΛN
v∈V :
(u,v)∈E
the last inequality follows by our choice of ǫ ′ . Hence, (3.28) holds. Consequently,
ρN H N ∈ [0, c3 δ|ΛN |/8] ≥ρN Uǫ′
≥(π(ǫ ′ )2 )|ΛN |−1 (2ǫ ′ )|ΛN | ≥ e c4 |ΛN |
(3.31)
with c4 = c4 (ǫ ′ ) = log(2π(ǫ ′ )3 ). Inserting this bound into (3.26) yields the claim of the lemma.
Lemma 3.4. Let c3 be as in Lemma 3.2 and c1 as in (2.6). For all δ > 0, β > 0, and N ∈ N, one has
«Œ
‚
¨
Z
3 4
1
2π
ǫ
β
c
δ
3
. (3.32)
+ log
H N e−β H N dρN ≤ (πǫ 2 )−1 exp |ΛN | 6c1 (ǫ) −
4
2
c3 β
c δ|Λ |/2<H <∞
3
N
N
Note that for large β, the leading term in the exponent in the claimed upper bound is −β c3 δ|ΛN |/4.
This is twice the leading part of the exponent in the lower bound (3.25) of the partition sum. The
comparison between these two leading terms plays an important role below in the estimate (3.37).
Proof of Lemma 3.4. Here is the rough idea. Using the key lower bound from Lemma 3.2, the integration over the α-variables is bounded by a gaussian integral. The remaining integration over the
x-variables is bounded by a power of the area of the disks associated to the constraints.
Recall that c1 (ǫ) < ∞ by (2.7). It follows from the definition of the Hamiltonian H N that
H N ≤ 6|ΛN |c1 (ǫ) ≤ exp(6|ΛN |c1 (ǫ)) holds on the set {H N < ∞}. Consequently, using exponential Chebyshev in the first step and applying Lemma 3.2 in the last step, we obtain
Z
H N e−β H N dρN
≤
Z
c3 δ|ΛN |/2<H N <∞
H N <∞
H N eβ(H N −c3 δ|ΛN |/2)/2 · e−β H N dρN
≤ exp({6c1 (ǫ) − β c3 δ/4}|ΛN |)
Z
≤ exp({6c1 (ǫ) − β c3 δ/4}|ΛN |)
Z
e−β H N /2 dρN
H N <∞
H N <∞
1718

exp −
c3 β X
2
u∈ΛN

α2u  dρN .
(3.33)
Next, we evaluate the last integral: Denote by B(x, r) the ball centered at x with radius r. Enumerate the vertices in ΛN as u1 , u2 , . . . , u|ΛN | in such a way that (ui , ui+1 ) ∈ E for all i and u|ΛN | = 0.
per
Consider ω ∈ ΩN with H N (ω) < ∞. By the description (3.3) of the set {H N < ∞}, for every
u ∈ ΛN , the component x u is contained in the intersection of the six balls B(x v + e iαu (u − v), ǫ),
where v runs through the neighbors of u. Consequently, to obtain an upper bound for the contribution to the integral coming from the integration over the x-variables, we can proceed as follows: we
integrate successively x ui , i = 1, 2, . . . , |ΛN | − 1 over B(x ui+1 + e iαui+1 (ui+1 − ui ), ǫ). Since x 0 = 0, the
integration over x u|Λ | yields no contribution. All other x u -integrations give as a factor the Lebesgue
N
measure of a ball of radius ǫ, namely πǫ 2 . Hence,


Z
Z
Y
c3 β X 2
c3 β 2
2 |ΛN |−1


exp −
exp −
dρN ≤(πǫ )
dαu
α
α
2 u∈Λ u
2 u
H N <∞
u∈ΛN R
N
2π |ΛN |/2
2 |ΛN |−1
≤(πǫ )
.
c3 β
(3.34)
Inserting this bound in (3.33) yields the claim of the lemma.
Proof of Theorem 2.3. Let δ > 0. First, we rewrite the variance of α0 using the properties of the law
of the αu ’s from Lemma 3.1, then we insert the bound from Lemma 3.2 for the Hamiltonian:
Varµβ,N (α0 ) =
=
1
X
|ΛN | u∈Λ
1
|ΛN |
Varµβ,N (αu )
N

Eµβ,N 
X
u∈ΛN

α2u 
≤
1
c3 |ΛN |
Eµβ,N [H N ].
(3.35)
To bound the last expectation, we split the domain of integration into two parts, using that H N < ∞
holds µβ,N -almost surely:
Eµβ,N [H N ] =Eµβ,N [H N 1{H N ≤c3 δ|ΛN |/2} ] +
1
Zβ,N
Z
H N e−β H N dρN .
(3.36)
c3 δ|ΛN |/2<H N <∞
The first summand is bounded by c3 δ|ΛN |/2. Denote the second summand in (3.36) by I. Inserting
the bound from Lemma 3.3 for some ǫ ′ ∈ (0, π−1/2 ) with c1 (3ǫ ′ ) < c3 δ/48, and the bound from
Lemma 3.4, we obtain:
‚
¨
«Œ
β c3 δ
β c3 δ 1
2π3 ǫ 4
2 −1
′
I ≤(πǫ ) exp |ΛN | −c4 (ǫ ) +
+ 6c1 (ǫ) −
+ log
8
4
2
c3 β
β c3 δ 1
− log(c3 β)
(3.37)
=(πǫ 2 )−1 exp |ΛN | c5 (ǫ ′ , ǫ) −
8
2
p
with c5 (ǫ ′ , ǫ) = −c4 (ǫ ′ ) + 6c1 (ǫ) + log 2π3 ǫ 4 .
Note that the leading contribution −|ΛN |β c3 δ/4 coming from Lemma 3.4 is only partially
1719
cancelled by the leading contribution |ΛN |β c3 δ/8 from Lemma 3.3.
max{1/c3 , 8c5 (ǫ ′ , ǫ)/(c3 δ)}, one obtains
I ≤(πǫ 2 )−1 ≤
c3
2
δ|ΛN |
For β > β0 :=
(3.38)
for all sufficiently large N , namely for N ≥ N0 := (c3 δπǫ 2 /2)−1/2 . Inserting this bound into (3.36)
yields
Eµβ,N [H N ] ≤ c3 δ|ΛN |.
(3.39)
Hence, (3.35) implies the claim Varµβ,N (α0 ) ≤ δ.
4 Infinite-volume arguments
In this section, we use tightness arguments to pass to the infinite-volume limit.
Lemma 4.1. The finite-dimensional marginals of µβ,N , N ∈ N, are tight. As a consequence, there exists
a strictly increasing sequence (Nk )k∈N of natural numbers such that the finite-dimensional marginals of
(µβ,Nk )k∈N converge weakly to the marginals of a limiting distribution µβ on ΩV .
Proof. By the definition of µβ,N , we have for any finite edge set F ⊂ E:
µβ,N ∀e = (u, v) ∈ F : |x u − x v − γu (u − v)| ≤ ǫ = 1.
(4.1)
Consequently, |x u − x v | ≤ 1 + ǫ holds µβ,N -almost surely for all (u, v) ∈ E. Since x 0 = 0, this implies
|x u | ≤ dist(u, 0)(1 + ǫ) for all u ∈ V µβ,N -almost surely, where dist(u, 0) denotes the graph distance
between u and 0 in the triangular lattice T . In particular, the finite-dimensional marginals of µβ,N
are tight. Thus, using a diagonal sequence argument, there exists a strictly increasing sequence
(Nk )k∈N of natural numbers such that the finite-dimensional marginals of µβ,Nk converge weakly to
the corresponding marginals of a probability measure µβ on ΩV .
In order to see that any infinite-volume limit µβ as in Lemma 4.1 is non-degenerate, we need to
show that no mass of the measures µβ,N can accumulate in the boundaries |x u − x v − γu (u − v)| = ǫ
of the admissible domain. More generally, the following lemma shows that no mass of the measures
µβ,N can accumulate on null sets with respect to the reference measure. The proof is based on an
entropy argument.
Lemma 4.2. For all finite connected Λ ⊂ V with 0 ∈ Λ and all β > 0, there exist constants c6 (β, ǫ) > 0
and c7 (Λ, ǫ) > 0 such that for all N large enough and all measurable A ⊆ ΩΛ with 0 < ρΛ [A] < c7 , one
has
µβ,N [ωΛ ∈ A] ≤
1720
c6
− log
ρΛ [A]
c7
.
(4.2)
Proof. Let us first explain the rough idea. The lower bound for the partition sum from Lemma 3.3
gives us an upper bound for the free energy per unit volume, and in turn an upper bound for the
entropy (with the mathematics sign convention) per unit volume. This is used to derive an upper
bound for the entropy of the system restricted to a fixed volume. Given this entropy bound, the
thermal measure cannot give too much mass to sets of small reference measure.
Fix a finite connected set Λ ⊂ V with 0 ∈ Λ, and let T be an (undirected) spanning tree of Λ. Let N
be so large that Λ is contained in the box ΛN . Take BN ,Λ to be a maximal subset of ΛN with 0 ∈ BN ,Λ
such that the sets b + Λ, b ∈ BN ,Λ , are pairwise disjoint subsets of ΛN . Note that there is a constant
c8 (Λ) > 0 such that for all large enough N one has
|BN ,Λ | ≥ c8 |ΛN |.
(4.3)
Let b + T denote the translation of the tree T by b. Furthermore, let TN be a spanning tree of ΛN
that contains all trees b + T , b ∈ BN ,Λ . Define
per
Ω̃N :={(x u , γu )u∈V ∈ ΩN : |x v − x u | ≤ 1 + ǫ for all (u, v) ∈ TN },
Ω̃Λ :={(x u , γu )u∈Λ ∈ ΩΛ : |x v − x u | ≤ 1 + ǫ for all (u, v) ∈ T };
(4.4)
(4.5)
per
here we have identified the trees TN and T with their edge sets. By (3.3), all configurations ω ∈ ΩN
with H N (ω) < ∞ are contained in the set Ω̃N . In particular, the thermal measure µβ,N is supported
on Ω̃N .
Recall the definition (2.4) of the reference measure ρN = ρΛN . In the remainder of this proof, we
work with the reference measure
ρ̃N (dω) :=
1Ω̃N (ω)
ρN (Ω̃N )
ρN (dω) =
1Ω̃N (ω)
(π(1 + ǫ)2 )|ΛN |−1 (2π)|ΛN |
ρN (dω).
(4.6)
The normalization is chosen in such a way that ρ̃N is a probability measure. Changing the normalization of the reference measure does not change the finite-volume Gibbs measure µβ,N , it changes
only the normalizing constant Zβ,N defined in (2.18). The new normalizing constant
Z
Z̃β,N :=
per
ΩN
e−β H N d ρ̃N
(4.7)
satisfies
Z̃β,N =
Zβ,N
(π(1 + ǫ)2 )|ΛN |−1 (2π)|ΛN |
≥
Zβ,N
(2π2 (1 + ǫ)2 )|ΛN |
.
(4.8)
By Lemma 3.3, there exists a constant c9 ∈ R such that Zβ,N ≥ exp(c9 |ΛN |) holds. (In the terminology of Lemma 3.3, we can use c9 = c4 − β c3 δ/8 for any admissible choice of δ and ǫ ′ .) Hence, it
follows that
Z̃β,N ≥ exp(−c10 |ΛN |)
(4.9)
with −c10 := c9 − log(2π2 (1 + ǫ)2 ).
By the definition (2.17) of µβ,N and the positivity of the Hamiltonian H N , we have
log
dµβ,N
d ρ̃N
= −β H N − log Z̃β,N ≤ − log Z̃β,N .
1721
(4.10)
We take the expectation in the last inequality and insert the bound (4.9) for the partition sum to
obtain the following estimate for the relative entropy:
™
–
dµβ,N
≤ − log Z̃β,N ≤ c10 |ΛN |.
(4.11)
Eµβ,N log
d ρ̃N
Since relative entropies between probability measures are non-negative, we know c10 ≥ 0.
Let λ denote the distribution of ωΛ with respect to ρ̃N , i.e.
λ(dωΛ ) =
1Ω̃Λ
c7
ρΛ (dωΛ )
(4.12)
with c7 = ρΛ [Ω̃Λ ] = (π(1 + ǫ)2 )|Λ|−1 (2π)|Λ| . For this proof, λ serves as a reference probability
measure on ΩΛ . We define
per
B
ψN : ΩN → ΩΛN ,Λ ,
(x u , γu )u∈V 7→ ((x b+u − x b , γ b+u )u∈Λ ) b∈BN ,Λ .
(4.13)
Πβ,N := ψN [µβ,N ]
(4.14)
Let
b
denote the image of µβ,N under ψN , and let Πβ,N
, b ∈ BN ,Λ , denote its marginals on ΩΛ . Furthermore, let
ζN := ψN [ρ̃N ]
(4.15)
be the image of the reference measure ρ̃N under ψN . Note that ζN = λBN ,Λ is a product measure.
The mathematical relative entropy can only decrease if we replace the measures by their image
measure with respect to ψN . Hence, we get
–
–
–
™
™
™
dΠβ,N
dψN [µβ,N ]
dµβ,N
≥EψN [µβ,N ] log
= EΠβ,N log
.
(4.16)
Eµβ,N log
d ρ̃N
dψN [ρ̃N ]
dζN
The last relative entropy can only decrease if we replace Πβ,N by the product measure Π̃β,N :=
Q
b
b∈BN ,Λ Πβ,N with the same marginals as Πβ,N . Since the distribution µβ,N is invariant under the
b
translation (x u , γu )u∈V 7→ (x u+b − x b , γu+b )u∈V for any b ∈ V , all marginals Πβ,N
of Πβ,N are equal.
Consequently,
–
™
™
–
dΠβ,N
d Π̃β,N
EΠβ,N log
≥EΠ̃β,N log
dζN
dζN




b
b
X
dΠβ,N
dΠβ,N
 = |BN ,Λ |E b log

(4.17)
=
EΠ b log
Πβ,N
β,N
dλ
dλ
b∈B
N ,Λ
for any b ∈ BN ,Λ . Combining the last inequality with (4.11) and (4.16), we obtain


b
dΠβ,N

c10 |ΛN | ≥ |BN ,Λ |EΠ b log
β,N
dλ
1722
(4.18)
for any b ∈ BN ,Λ . Hence, by (4.3), we get
c10
c8
for all sufficiently large N .

≥ EΠ b log
β,N
b
dΠβ,N
dλ

(4.19)

Fix a measurable set A ∈ ΩΛ with 0 < λ(A) < 1. Since the relative entropy can only decrease if we
take the image measure with respect to the indicator function 1A of the set A, it follows that




b
b
d1A[Πβ,N
]
dΠβ,N

≥E

EΠ b log
b
1A[Πβ,N
] log
β,N
dλ
d1A[λ]
=
b
[A] log
Πβ,N
b
[A]
Πβ,N
λ[A]
b
[Ac ] log
+ Πβ,N
b
[Ac ]
Πβ,N
λ[Ac ]
2
2
b
b
b
≥ − − Πβ,N
[Ac ] log λ[Ac ] ≥ − − Πβ,N
[A] log λ[A] − Πβ,N
[A] log λ[A];
e
e
(4.20)
here we used that x log x ≥ −1/e for all x > 0, and λ[Ac ] ≤ 1, since λ is a probability measure.
Combining (4.19) and (4.20), we obtain the following inequality:
b
[A] ≤
µβ,N [ωΛ ∈ A] = Πβ,N
c6
− log λ[A]
=
c6
− log
ρΛ [A]
c7
(4.21)
with c6 = c10 /c8 + 2/e; note that − log λ[A] > 0.
Corollary 4.3. Let µβ be an infinite-volume limiting distribution as in Lemma 4.1.
(a) The finite-dimensional marginals µβ [ωΛ ∈ ·] are absolutely continuous with respect to the reference measures ρΛ .
(b) For all (u, v) ∈ E, the bound |x u − x v − γu (u − v)| < ǫ holds µβ -almost surely.
(c) The local partition sums Zβ,Λ (ω∂ Λ ) arising in (2.11) are strictly positive for µβ -almost all ω.
Proof. (a) Let N ⊂ ΩΛ be a null set with respect to ρΛ . Then for every δ > 0, there is an open set
Aδ ⊆ ΩΛ with respect to the natural topology on ΩΛ such that N ⊆ Aδ and 0 < ρΛ [Aδ ] ≤ δ. Since
the marginals of a subsequence (µβ,Nk )k∈N converge weakly to the corresponding marginals of µβ
by Lemma 4.1, we conclude for all small δ, using the bound (4.2):
µβ [ωΛ ∈ N ] ≤ µβ [ωΛ ∈ Aδ ] ≤ lim inf µβ,Nk [ωΛ ∈ Aδ ]
k→∞
≤
c6
− log
ρΛ [Aδ ]
c7
≤
c6
− log
δ↓0
δ
c7
−→ 0.
(4.22)
Thus, µβ [ωΛ ∈ N ] = 0.
(b) We note that for all (u, v) ∈ E, the weaker bound |x u −x v −γu (u−v)| ≤ ǫ holds µβ,N -almost surely
for all N . Since the set of all (x u , γu , x v , γ v ) fulfilling this bound is closed, the same bound holds also
µβ -almost surely. Furthermore, the set of all (x u , γu , x v , γ v ) fulfilling |x u − x v − γu (u − v)| = ǫ is a
1723
null set with respect to ρ{u,v} . Using part (a), this set is also a null set with respect to the marginal
of the measure µβ on Ω{u,v} . Together, this implies part (b).
Part (c) is an immediate consequence of part (b), since Zβ,Λ (ω∂ Λ ) is strictly positive for all ω ∈ ΩV
with |x u − x v − γu (u − v)| < ǫ for all (u, v) ∈ E.
Proof of Theorem 2.4. Fix β > 0, a finite set Λ ⊂ V , and take N ∈ N large enough. We know the
DLR conditions (2.11) for the finite-volume Gibbs measures µβ,N instead of µβ . We rewrite these
conditions in integral form as follows: For every finite Σ ⊂ V with Λ ∪ ∂ Λ ⊆ Σ, every bounded and
continuous function f : ΩΣ → R, and every sufficiently large N , writing µβ,N ,Σ for the marginal of
µβ,N on ΩΣ , one has
Z
Zβ,Λ (χ∂ Λ ) f (χΣ ) µβ,N ,Σ (dχΣ )
ΩΣ
=
Z
ΩΣ
Z
f (ωΛ χΣ\Λ )e−β HΛ (ωΛ |χ∂ Λ ) ρΛ (dωΛ ) µβ,N ,Σ (dχΣ ).
(4.23)
ΩΛ
Substituting the definition (2.9) of Zβ,Λ (χ∂ Λ ) and taking the difference of both sides, this is equivalent to
Z Z
ΩΣ
ΩΛ
[ f (χΣ ) − f (ωΛ χΣ\Λ )]e−β HΛ (ωΛ |χ∂ Λ ) ρΛ (dωΛ ) µβ,N ,Σ (dχΣ ) = 0.
(4.24)
Now, since the potential h is continuous almost everywhere on the set {a ∈ C : |a| < ǫ}, the
integrand in (4.24)
ΩΛ × ΩΣ ∋ (ωΛ , χΣ ) 7→ [ f (χΣ ) − f (ωΛ χΣ\Λ )]e−β HΛ (ωΛ |χ∂ Λ )
(4.25)
is almost everywhere continuous with respect to ρΛ × ρΣ . Writing µβ,Σ for the marginal of µβ on
ΩΣ , part (a) of Corollary 4.3 implies that ρΛ ×µβ,Σ is absolutely continuous with respect to ρΛ ×ρΣ .
Hence, the function in (4.25) is also almost everywhere continuous with respect to ρΛ × µβ,Σ . As
this function is bounded, we can pass to the infinite-volume limit in (4.24):
Z Z
ΩΣ
ΩΛ
[ f (χΣ ) − f (ωΛ χΣ\Λ )]e−β HΛ (ωΛ |χ∂ Λ ) ρΛ (dωΛ ) µβ,Σ (dχΣ ) = 0.
(4.26)
This identity together with the positivity of the local partition sums (part (c) of Corollary 4.3) is
equivalent to the DLR condition (2.11).
Proof of Theorem 2.1. By Theorem 2.4, the µβ,Nk -distribution of γ0 = e iα0 converges weakly as k →
∞ to the µβ -distribution of γ0 . Since e iα 7→ α ∈ (−π, π] is discontinuous only at the point e iα = −1
and µβ [γ0 = −1] = 0, Lemma 3.1 implies
Eµβ [α0 ] = lim Eµβ,N [α0 ] = 0.
k→∞
k
(4.27)
Using Theorem 2.3, we conclude that for δ > 0, we have
lim Varµβ (α0 ) = lim lim Varµβ ,Nk (α0 ) ≤ δ.
β→∞
β→∞ k→∞
Hence, it follows that limβ→∞ Varµβ (α0 ) = 0.
1724
(4.28)
Sketch of proof of Remark 2.2. Given ǫ, Λ, and ϕ as in Remark 2.2, we fix R > maxu∈Λ |u|, and take
a smooth function f : [0, ∞) → [0, 1] with f (r) = 1 for 0 ≤ r ≤ R, f (r) = 0 for all large r, and
sup r>0 (r + 1)| f ′ (r)| < ǫ/ϕ. Taking log r times a negative constant close to 0, and translating and
truncating this function appropriately, one can see that such a function f exists. We set for u ∈ V
and −ϕ ≤ t ≤ ϕ
x u (t) = e i t f (|u|) u,
γu (t) = e i t f (|u|)
(4.29)
For this choice, the properties (a), (c), (d), and (e) are obviously true. To see the bound in (b), we
estimate for t ∈ [−ϕ, ϕ] and (u, v) ∈ E, for some value r between |u| and |v|, using ||u| − |v|| ≤ 1:
|x u (t) − x v (t) − γu (t)(u − v)| = |v| · |e i t f (|u|) − e i t f (|v|) | ≤ |v| · |t| · | f (|u|) − f (|v|)|
≤ ϕ · |v| · | f ′ (r)| ≤ ϕ · (r + 1)| f ′ (r)| < ǫ.
(4.30)
Acknowledgement We would like to thank two anonymous referees for making useful suggestions.
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